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E-polynomial of SL₂(ℂ)-character varieties of free groups

Samuel Cavazos

Mathematics Department, Northwestern University, 2033 Sheridan Road Evanston, IL 60208-2730, USA

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and

Sean Lawton

Department of Mathematics, The University of Texas-Pan American, 1201 West University Drive Edinburg, TX 78539, USA

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https://doi.org/10.1142/S0129167X1450058XCited by:8 (Source: Crossref)

Abstract

Let 𝖥_r be a free group of rank r, 𝔽_q a finite field of order q, and let SL_n(𝔽_q) act on Hom(𝖥_r, SL_n(𝔽_q)) by conjugation. We describe a general algorithm to determine the cardinality of the set of orbits Hom(𝖥_r, SL_n(𝔽_q))/SL_n(𝔽_q). Our first main theorem is the implementation of this algorithm in the case n = 2. As an application, we determine the E-polynomial of the character variety Hom(𝖥_r, SL₂(ℂ))//SL₂(ℂ), and of its smooth and singular locus. Thus we determine the Euler characteristic of these spaces.

Dedicated to Adalyn Belle Cavazos

Keywords:

AMSC: 14L30, 14D20, 14G05, 14G15

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